Exponents & Scientific Notation MATH 102 Contemporary Math S. Rook

Overview Section 6.5 in the textbook: – Exponent rules – Scientific notation

Exponent Rules

4 4 Review of Exponential Notation Consider (-3) 4 What is its expanded form? What about -3 4 ? Recall an exponential expression is made up of a base raised to a power x a = x · x · x · x · x · x · x · … · x (a times) – Identifying the base is the key

5 5 Product Rule Consider 2 2 ∙ 2 4 How does this expand? Product Rule: x a ∙ x b = x a+b – When multiplying LIKE BASES (the same variable), add the exponents – Only applies when the operation is multiplication

6 6 Power Rule Consider (2 2 ) 4 How does this expand? Power Rule: (x a ) b = x ab – When raising variables to a power, multiply the exponents – Only applies when the exponent is outside a set of parentheses

7 7 PRODUCT Rule versus POWER Rule Be careful not to confuse: – Product Rule: x 4 · x 7 (multiplying LIKE bases) – Power Rule: (x 4 ) 7 (exponent appears with NO base) – It is a common mistake to mix up the Product Rule and the Power Rule!

8 8 Quotient Rule Consider 3 5 / 3 2 How does this expand? Quotient Rule: x a / x b = x a-b – When dividing LIKE BASES (the same variable), subtract the exponents – Only applies when the operation is division

Exponent Rules (Example) Ex 1: Use the exponent rules to evaluate: a) 3 2 x 3 4 b) (2 3 ) 2 c) 5 9 / 5 7

10 Expressions with Negative Exponents Consider 2 2 / by the quotient rule Usually, we do NOT leave an expression with a negative exponent Flipping an exponent AND its base from the numerator into the denominator (or vice versa) reverses the sign of the exponent – e.g = 1 / 3 2

11 Expressions with Negative Exponents (Continued) How would we evaluate 2 -3 ? 2 -3 ≠ -8 – The sign of the exponent DOES NOT affect the sign of the base! – Whenever using the quotient rule, the result goes into the numerator

Exponent Rules (Example) Ex 2: Use the exponent rules to evaluate: a) 2 -4 x 2 2 b) (3 3 ) -2 c) 6 -2 / 6 -4

Scientific Notation

14 Writing Numbers in Scientific Notation Scientific Notation: any number in the form of a x 10 b where -10 < a < 10, a ≠ 0 and b is an integer – Used to write extreme numbers (large or small) in a compact format To write a number in scientific notation: – Place the decimal point so that one non-zero number is to the left of the decimal point and the rest of the numbers are to the right

Writing Numbers in Scientific Notation (Continued) – Determine the effect of moving the decimal point: Count how many places the decimal point is moved If the original number (without the sign) is greater than 1, b (the exponent) is positive If the original number (without the sign) is less than 1, b is negative 15

Scientific Notation (Example) Ex 3: Rewrite in scientific notation: a) 4,356,000 b) 0.008

17 Scientific Notation to Standard Form Standard Notation: writing a number expressed in scientific notation without the power of ten – To convert to standard notation, take the decimal and move it: To the right if b (the exponent) is positive To the left if b (the exponent) is negative Fill in empty spots with zeros

Scientific Notation (Example) Ex 4: Rewrite in standard notation: a) 4.5 x b) 3.25 x 10 4

19 Multiplying or Dividing in Scientific Notation Multiply or divide the numbers as normal Use the Product or Quotient Rules to simplify the power of tens Write the final answer in scientific notation

Scientific Notation (Example) Ex 5: Perform the following operations and leave the answer in scientific notation: a) (1.2 x )(3 x 10 5 ) b) (4.8 x 10 4 ) / (1.6 x ) c)

Summary After studying these slides, you should know how to do the following: – Apply the exponent rules to any numerical base – Convert from standard notation to scientific notation and vice versa – Multiply and divide numbers in scientific notation Additional Practice: – See problems in Section 6.5 Next Lesson: – Sequences (Section 6.6)